Isaac Newton's notation for differentiation (also called the
dot notation,
fluxions, or sometimes, crudely, the
flyspeck notation for differentiation) places a dot over the dependent variable. That is, if
y is a function of
t, then the derivative of
y with respect to
t is :\dot y Higher derivatives are represented using multiple dots, as in :\ddot y, \overset{...}{y} Newton extended this idea quite far: :\begin{align} \ddot{y} &\equiv \frac{d^2y}{dt^2} = \frac{d}{dt}\left(\frac{dy}{dt}\right) = \frac{d}{dt}\Bigl(\dot{y}\Bigr) = \frac{d}{dt}\Bigl(f'(t)\Bigr) = D_t^2 y = f
(t) = y_t \\[5pt] \overset{...}{y} &= \dot{\ddot{y}} \equiv \frac{d^3y}{dt^3} = D_t^3 y = f
(t) = y_t \\[5pt] \overset{\,4}{\dot{y}} &= \overset{....}{y} = \ddot{\ddot{y}} \equiv \frac{d^4y}{dt^4} = D_t^4 y = f^{\rm IV}(t) = y^{(4)}_t \\[5pt] \overset{\,5}{\dot{y}} &= \ddot{\overset{...}{y}} = \dot{\ddot{\ddot{y}}} = \ddot{\dot{\ddot{y}}} \equiv \frac{d^5y}{dt^5} = D_t^5 y = f^{\rm V}(t) = y^{(5)}_t \\[5pt] \overset{\,6}{\dot{y}} &= \overset{...}{\overset{...}{y}} \equiv \frac{d^6y}{dt^6} = D_t^6 y = f^{\rm VI}(t) = y^{(6)}_t \\[5pt] \overset{\,7}{\dot{y}} &= \dot{\overset{...}{\overset{...}{y}}} \equiv \frac{d^7y}{dt^7} = D_t^7 y = f^{\rm VII}(t) = y^{(7)}_t \\[5pt] \overset{\,10}{\dot{y}} &= \ddot{\ddot{\ddot{\ddot{\ddot{y}}}}} \equiv \frac{d^{10}y}{dt^{10}} = D_t^{10} y = f^{\rm X}(t) = y^{(10)}_t \\[5pt] \overset{\,n}{\dot{y}} &\equiv \frac{d^ny}{dt^n} = D_t^n y = f^{(n)}(t) = y^{(n)}_t \end{align} Unicode characters related to Newton's notation include: • • • ← replaced by "combining diaeresis" + "combining dot above". • ← replaced by "combining diaeresis" twice. • • • • • Newton's notation is generally used when the independent variable denotes
time. If location is a function of
t, then \dot y denotes
velocity and \ddot y denotes
acceleration. This notation is popular in
physics and
mathematical physics. It also appears in areas of mathematics connected with physics such as
differential equations. When taking the derivative of a dependent variable
y =
f(
x), an alternative notation exists: :\frac{\dot{y}}{\dot{x}} = \dot{y}:\dot{x} \equiv \frac{dy}{dt}:\frac{dx}{dt} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} = \frac{dy}{dx} = \frac{d}{dx}\Bigl(f(x)\Bigr) = D y = f'(x) = y'. Newton developed the following partial differential operators using side-dots on a curved X ( ⵋ ). Definitions given by Whiteside are below: : \begin{align} \mathcal{X} \ &=\ f(x,y) \,, \\[5pt] \cdot\mathcal{X} \ &=\ x\frac{\partial f}{\partial x} = xf_x\,, \\[5pt] \mathcal{X}\!\cdot \ &=\ y\frac{\partial f}{\partial y} = yf_y\,, \\[5pt] \colon\!\mathcal{X}\,\text{ or }\,\cdot\!\left(\cdot\mathcal{X}\right) \ &=\ x^2\frac{\partial^2 f}{\partial x^2} = x^2 f_{xx}\,, \\[5pt] \mathcal{X}\colon\,\text{ or }\,\left(\mathcal{X}\cdot\right)\!\cdot \ &=\ y^2\frac{\partial^2 f}{\partial y^2} = y^2 f_{yy}\,, \\[5pt] \cdot\mathcal{X}\!\cdot\ \ &=\ xy\frac{\partial^2 f}{\partial x \, \partial y} = xy f_{xy}\,, \end{align}
Newton's notation for integration Newton developed many different notations for
integration in his
Quadratura curvarum (1704) and
later works: he wrote a small vertical bar or prime above the dependent variable ( ), a prefixing rectangle (), or the inclosure of the term in a rectangle () to denote the
fluent or time integral (
absement). : \begin{align} y &= \Box \dot{y} \equiv \int \dot{y} \,dt = \int f'(t) \,dt = D_t^{-1} (D_t y) = f(t) + C_0 = y_t + C_0 \\ \overset{\,\prime}{y} &= \Box y \equiv \int y \,dt = \int f(t) \,dt = D_t^{-1} y = F(t) + C_1 \end{align} To denote multiple integrals, Newton used two small vertical bars or primes (), or a combination of previous symbols , to denote the second time integral (absity). : \overset{\,\prime\prime}{y} = \Box \overset{\,\prime}{y} \equiv \int \overset{\,\prime}{y} \,dt = \int F(t) \,dt = D_t^{-2} y = g(t) + C_2 Higher order time integrals were as follows: : \begin{align} \overset{\,\prime\prime\prime}{y} &= \Box \overset{\,\prime\prime}{y} \equiv \int \overset{\,\prime\prime}{y} \,dt = \int g(t) \,dt = D_t^{-3} y = G(t) + C_3 \\ \overset{\,\prime\prime\prime\prime}{y} &= \Box \overset{\,\prime\prime\prime}{y} \equiv \int \overset{\,\prime\prime\prime}{y} \,dt = \int G(t) \,dt = D_t^{-4} y = h(t) + C_4 \\ \overset{\;n}\overset{\,\prime}{y} &= \Box \overset{\;n-1}\overset{\,\prime}y \equiv \int \overset{\;n-1}\overset{\,\prime}y \,dt = \int s(t) \,dt = D_t^{-n} y = S(t) + C_n \end{align} This
mathematical notation did not become widespread because of printing difficulties and the
Leibniz–Newton calculus controversy. == Partial derivatives ==